Beam Deflection Formulas for Common Load Cases

All closed-form beam deflection formulas come from integrating the Euler-Bernoulli differential equation EI·d²y/dx² = M(x) twice along the beam span with the boundary conditions that define the support type. The product EI — flexural rigidity — is the single most important quantity: E is the material's Young's modulus (how stiff the material is) and I is the second moment of area of the cross-section about the bending axis (how efficiently the material is distributed away from the neutral axis). Doubling the span increases deflection by 2³ = 8× for a point load; doubling EI halves it. Choosing a deeper section rather than a heavier one is almost always the correct trade-off, because I scales with depth cubed.

The second moment of area for standard cross-sections follows directly from geometry: for a solid circular bar of diameter d, I = π·d⁴/64; for a solid rectangle of width b and depth h, I = b·h³/12; for a hollow tube, subtract the inner section's I from the outer. I-sections are efficient in bending precisely because material is concentrated in the flanges, far from the neutral axis, where it contributes most to I.

The table below gives the maximum deflection for each of the six load cases supported by the calculator. All formulas produce deflection in the same units as L (typically mm) when F is in N, w is in N/mm, E is in N/mm² and I is in mm⁴. The location of the maximum deflection is shown alongside each formula.

Simply supported beam, central point load: the maximum deflection occurs at midspan and is δ_max = F·L³/(48·E·I). This is the most-used case in structural checking. Simply supported beam, uniformly distributed load (UDL): the maximum deflection at midspan is δ_max = 5·w·L⁴/(384·E·I). The factor 5/384 ≈ 0.01302 is commonly memorised. Cantilever beam, end point load: the tip deflects by δ_max = F·L³/(3·E·I) — sixteen times more than the simply supported case for the same span, load and section, because the effective span is doubled and the boundary condition halves the restraint. Cantilever, UDL: the tip deflection is δ_max = w·L⁴/(8·E·I). Fixed-fixed beam, central point load: end fixity reduces the midspan deflection to δ_max = F·L³/(192·E·I) — four times stiffer than simply supported — but requires that the supports can develop the end moments M_A = M_B = F·L/8.

Once the peak bending moment M_max is known from the load case, the maximum bending stress follows from the flexure formula σ = M·c/I, where c is the distance from the neutral axis to the outermost fibre of the cross-section (half the depth for a symmetric section). The bending stress must not exceed the yield strength Sy of the material; in practice a safety factor SF = Sy/σ_max of at least 1.5–2 is expected for most structural applications, and the governing code (EN 1993-1-1 for steel) sets the partial factor to apply.

For the simply supported central-load case, the peak bending moment at midspan is M_max = F·L/4. For a cantilever with an end load it is M_max = F·L at the root. For a UDL on a simply supported beam, M_max = w·L²/8 at midspan. These moment values are inputs to the stress check — the deflection check and the stress check are both required and are independent of each other.

The two things to get right before applying any formula are the support condition and the load type. A 'simply supported' beam is one that is free to rotate at both ends but cannot translate vertically; this is a close model for a beam resting on two knife-edge supports, standard shelf bearings or pinned-end connections. A 'fixed' or 'clamped' end cannot rotate and cannot translate — this requires a moment-resisting connection such as a welded end plate or embedded foundation. In practice, most real connections are somewhere in between, so assuming simply supported is often the more conservative (larger deflection) assumption unless the fixity is deliberate and verified.

For spans carrying both a point load and a uniform distributed load simultaneously, the deflections and bending moments from the two cases may be added directly (superposition), because Euler-Bernoulli beam theory is linear. The MechanixCalc combined-loading panel applies superposition automatically for SS and cantilever beams. For two-span continuous beams the calculator applies the 3-moment (Clapeyron) equation, which accounts for the bending moment at the interior support coupling the two spans together.